Lemma 3 
Let \(\Phi : \mathcal {A}\rightarrow \mathcal {B}(\mathcal {K})\)  be a \(C^*\) -homomorphism. Then \(\Phi ^*: \mathcal {A}\rightarrow \mathcal {B}(\mathcal {K})\)  is a \(C^*\) -homomorphism if and only if \(\Phi (\mathcal {A}) \subset \mathcal {B}(\mathcal {K})\)  is a commutative subalgebra,It is interesting to note that Bell's theorem holds for states over \(C^*\) -algebras as long as one of them is commutative [1]}. We discuss the relation with Bell's theorem and Bell nonlocality elsewhere [2]}. equivalently, \(\Phi = \Phi |_V\)  for \(V \subset \mathcal {A}\)  a commutative subalgebra.

